Problem: Simplify; express your answer in exponential form. Assume $a\neq 0, z\neq 0$. $\dfrac{{(a^{4})^{5}}}{{(a^{4}z^{-2})^{-2}}}$
Solution: To start, try working on the numerator and the denominator independently. In the numerator, we have ${a^{4}}$ to the exponent ${5}$ . Now ${4 \times 5 = 20}$ , so ${(a^{4})^{5} = a^{20}}$ In the denominator, we can use the distributive property of exponents. ${(a^{4}z^{-2})^{-2} = (a^{4})^{-2}(z^{-2})^{-2}}$ Simplify using the same method from the numerator and put the entire equation together. $\dfrac{{(a^{4})^{5}}}{{(a^{4}z^{-2})^{-2}}} = \dfrac{{a^{20}}}{{a^{-8}z^{4}}}$ Break up the equation by variable and simplify. $\dfrac{{a^{20}}}{{a^{-8}z^{4}}} = \dfrac{{a^{20}}}{{a^{-8}}} \cdot \dfrac{{1}}{{z^{4}}} = a^{{20} - {(-8)}} \cdot z^{- {4}} = a^{28}z^{-4}$.